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Journal of Optics

Quantum field theory of single-photon states

Andrea Aiello

Published May 23, 2025• 3 citations
Publisher

Abstract

Abstract We present a detailed quantum field theory of single-photon states. The formalism we develop permits us to calculate the electric field of a single-photon light beam sent through a beam splitter (BS). This illustrates the nonlocal nature of quantum fields even when they are excited by discrete numbers of quanta. We find that the shape of the electromagnetic wave attached to a single photon entering the BS, determines the results of simultaneous measurements at two different locations behind the two ports of the BS. Therefore, our results confirm that a quantum field can be both local with respect to its discrete excitations, i.e. the photons, and nonlocal with respect to its continuous amplitude.

DOI
10.1088/2040-8986/addc7b

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Abstract We present a detailed quantum field theory of single-photon states. The formalism we develop permits us to calculate the electric field of a single-photon light beam sent through a beam splitter (BS). This illustrates the nonlocal nature of quantum fields even when they are excited by discrete numbers of quanta. We find that the shape of the electromagnetic wave attached to a single photon entering the BS, determines the results of simultaneous measurements at two different locations behind the two ports of the BS. Therefore, our results confirm that a quantum field can be both local with respect to its discrete excitations, i.e. the photons, and nonlocal with respect to its continuous amplitude.
It is well known that the expectation value of the electric field operator in a number (Fock) state vanishes. However, a measurement of the electric field must yield some nonzero value each time an experiment is performed. What values can be obtained, and with what probability, is perhaps less well known. In this work we determine the probability distribution for the electric-field amplitude of an optical beam prepared in a single-photon state, using some techniques of quantum field theory. For the purpose of illustration, we apply our theory to the analysis of an ideal experiment where two spatially separated detectors can measure either the amplitude of the electric field, or the intensity of a single-photon optical beam. We find that while the photon cannot be detected in two disjoint positions at the same time, a measurement of the amplitude of the electric field carried by ditto photon, would yield a nonzero value at both detectors. These results show that a quantum field is both local with respect to its discrete excitations, i.e., the particles, and nonlocal with respect to its continuous amplitude.
We present a simple yet rigorous field theoretic demonstration of the nonlocality of a single-photon field. The formalism used allows us to calculate the electric field of a single-photon light beam sent through a beam splitter, which directly demonstrates that it is the light field, rather than the photon itself regarded as a particle, that exhibits nonlocality. Our results are obtained without using either inequalities or specific measurement apparatuses, so that they have perfectly general validity.
The concept of the photon is introduced by discussion of the process of electromagnetic field quantization within a closed cavity or in an open optical system. The nature of a single-photon state is clarified by consideration of its behaviour at an optical beam splitter. The importance of linear superposition or entangled states in the distinctions between quantum-mechanical photon states and classical excitations of the electromagnetic field is emphasized. These concepts and the ideas of wave-particle duality are illustrated by discussions of the effects of single-photon inputs to Brown-Twiss and Mach-Zehnder interferometers. Both the theoretical predictions and the-confirming experimental observations are covered. The defining property of the single photon in terms of its ability to trigger one. and only one, photodetection event is discussed.
Abstract The theory of the electromagnetic field quantization developed in Chapter 4 simplifies considerably for applications Lo many of the key experiments in quantum optics. Thus many experiments use plane parallel light beams whose transverse intensity profiles are not important for the measured quantities. Again, it is often sufficient in interpreting the data to consider the light beams as exciting a single mode of the field. The present chapter begins with the specialization of the quantization to single-mode excitations. The quantum theory is illustrated by its application to several varieties of single-mode state and the main features of number Cates, coherent states, chaotic statistical mixtures and squeezed states are derived.
Abstract Photons are the elementary quantum excitations of the electromagnetic field. Quantization is usually constructed on the basis of an expansion in eigenmodes, in the form of plane waves. Since they form a basis, other electromagnetic configurations can be constructed by linear combinations. In this presentation we discuss a formulation constructed in the general formalism of bosonic Fock space, in which the quantum excitation can be constructed directly on localized pulses of arbitrary shape. Although the two formulations are essentially equivalent, the direct formulation in terms of pulses has some conceptual and practical advantages, which we illustrate with some examples. The first one is the passage of a single photon pulse through a beam splitter. The analysis of this formulation in terms of pulses in Fock space shows that there is no need to introduce ‘vacuum fluctuations entering through the unused port’, as is often done in the literature. Another example is the Hong–Ou–Mandel effect. It is described as a time dependent process in the Schrödinger representation in Fock space. The analysis shows explicitly how the two essential ingredients of the Hong–Ou–Mandel effect are the same shape of the pulses and the bosonic nature of photons. This formulation shows that all the phenomena involving linear quantum optical devices can be described and calculated on the basis of the time dependent solution of the corresponding classical Maxwell’s equations for pulses, from which the quantum dynamics in Fock space can be immediately constructed.
One of the most puzzling consequences of interpreting quantum mechanics in terms of concepts borrowed from classical physics, is the so-called wave-particle duality. Usually, wave-particle duality is illustrated in terms of complementarity between path distinguishability and fringe visibility in interference experiments. In this work, we instead propose a new type of complementarity, that between the continuous nature of waves and the discrete character of particles. Using the probabilistic methods of quantum field theory, we show that the simultaneous measurement of the wave amplitude and the number of photons in the same beam of light is, under certain circumstances, prohibited by the laws of quantum mechanics. Our results suggest that the concept of “interferometric duality'' could be eventually replaced by the more general one of “continuous-discrete duality''.